E is for Evidence
Abstract. How much evidence do the data give us about one hypothesis versus another? The standard way to measure evidence is still the p-value, despite a myriad of problems surrounding it. We present the e-value, a recently popularized notion of evidence which overcomes some of these issues. While e-values were only given a name as recently as 2019, interest in them has since exploded with papers in the Annals, JRSS B, Biometrika and the like – June 2022 saw the first international workshop on e-values, a second one is planned.
In simple cases, e-values coincide with Bayes factors. But if the null is composite or nonparametric, or an alternative cannot be explicitly formulated, e-values and Bayes factors become distinct and e-processes can be seen as a generalization of nonnegative supermartingales. Thus, unlike the Bayes factor, e-values always allow for tests with strict frequentist Type-I error control under optional continuation of data collection and combination of data from different sources. E-values are also the basic building blocks of anytime-valid confidence intervals that remain valid under continuous monitoring and optional stopping. In parametric settings they tend to be strictly wider than, hence consistent with Bayesian credible intervals. This led to the development of the e-posterior, an analogue to the Bayesian posterior that gets wider rather than wrong if the prior is chosen badly.
This work is based on:
P. Grunwald,. R. de Heide, W. Koolen (2023). Safe Testing. To appear in J. Roy. Stat. Soc., Series B
P. Grunwald (2023) . The E-Posterior. Proc. Phil. Trans. Soc. London Series A, 2023.